Another Laurent Expansion Question

In summary, the conversation was about finding all possible Laurent expansions centered at 0 for (z - 1) / (z + 1) and determining the largest open set on which it converges. The attempt at a solution involved breaking down the expression into two parts, [z / (z+1)] and [1 / (z+1)], and trying to simplify each part individually. However, there were some uncertainties about the process and the possibility of having a negative series. There was also a question about how to handle the expression -1 / (z + 1).
  • #1
brianhawaiian
12
0

Homework Statement


Find all possible Laurent expansions centered at 0 for
(z - 1) / (z + 1)

Find the Laurent Expansion centerd at z = -1 that converages at z = 1/2 and determine the largest opens et on which
(z - 1) / (z + 1) converges



Homework Equations





The Attempt at a Solution



(z - 1) / (z + 1) breaks down into [z / (z+1)] - [1 / (z+1)]

For the first one divide out the z to obtain 1 / 1 + (1/z) I think? However not being in the form 1 / 1 - (1/z) would this be the same series but negative? That doens't seem right.

For breaking down -1 / (z + 1) I didn't know how to attack that one.
 
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  • #2
Anything?
 

1. What is a Laurent expansion?

A Laurent expansion is a mathematical series representation of a function that includes both positive and negative powers of the variable. It is used to express functions that have singularities, such as poles or branch points.

2. How is a Laurent expansion different from a Taylor series?

A Taylor series only includes non-negative powers of the variable, while a Laurent expansion includes both positive and negative powers. Additionally, a Taylor series is centered around a point, while a Laurent expansion is centered around a singularity.

3. What are the applications of Laurent expansions?

Laurent expansions have various applications in mathematics, physics, and engineering. They are used to solve differential equations, evaluate complex integrals, and analyze functions with singularities.

4. How do you find the coefficients in a Laurent expansion?

The coefficients in a Laurent expansion can be found by using the Cauchy integral formula or by using the residue theorem. These methods involve contour integration and complex analysis techniques.

5. Can any function be represented by a Laurent expansion?

No, not all functions can be represented by a Laurent expansion. Only functions that have singularities, such as poles or branch points, can be expressed as a Laurent series.

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